The equivalence between elliptic curve and quadratic function field discrete logarithms in characteristic 2

1998 ◽  
pp. 621-638 ◽  
Author(s):  
Robert J. Zuccherato
Keyword(s):  
Elliptic Curve ◽  
Function Field ◽  
Quadratic Function ◽  
Discrete Logarithms ◽  
Characteristic 2 ◽  
Quadratic Function Field

Class Number Divisibility in Real Quadratic Function Fields

1992 ◽  
Vol 35 (3) ◽  
pp. 361-370 ◽  
Author(s):  
Christian Friesen
Keyword(s):  
Class Number ◽  
Function Field ◽  
Class Group ◽  
Quadratic Function ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Quadratic Function Field ◽  
Quadratic Function Fields ◽  
Real Quadratic ◽  
The Ideal

AbstractLet q be a positive power of an odd prime p, and let Fq(t) be the function field with coefficients in the finite field of q elements. Let denote the ideal class number of the real quadratic function field obtained by adjoining the square root of an even-degree monic . The following theorem is proved: Let n ≧ 1 be an integer not divisible by p. Then there exist infinitely many monic, squarefree polynomials, such that n divides the class number, . The proof constructs an element of order n in the ideal class group.

scholarly journals Pseudo-elliptic integrals, units, and torsion

2005 ◽  
Vol 79 (3) ◽  
pp. 335-347 ◽  
Author(s):  
Francesco Pappalardi ◽  
Alfred J. Van Der Poorten
Keyword(s):  
Function Field ◽  
Function Fields ◽  
Quadratic Function ◽  
Elliptic Integrals ◽  
Square Root ◽  
Base Field ◽  
Quadratic Function Field

AbstractWe remark on pseudo-elliptic integrals and on exceptional function fields, namely function fields defined over an infinite base field but nonetheless containing non-trivial units. Our emphasis is on some elementary criteria that must be satisfied by a squarefree polynomialD(x)whose square root generates a quadratic function field with non-trivial unit. We detail the genus I case.

scholarly journals Elliptic Curves over the Perfect Closure of a Function Field

2010 ◽  
Vol 53 (1) ◽  
pp. 87-94
Author(s):  
Dragos Ghioca
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Function Field ◽  
Positive Characteristic ◽  
Rational Points ◽  
Finitely Generated

AbstractWe prove that the group of rational points of a non-isotrivial elliptic curve defined over the perfect closure of a function field in positive characteristic is finitely generated.

scholarly journals Points of low height on elliptic surfaces with torsion

2010 ◽  
Vol 13 ◽  
pp. 370-387
Author(s):  
Sonal Jain
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Open Subset ◽  
Function Field ◽  
Rational Point ◽  
Torsion Point ◽  
Integral Points ◽  
Elliptic Surfaces ◽  
Canonical Height

AbstractWe determine the smallest possible canonical height$\hat {h}(P)$for a non-torsion pointPof an elliptic curveEover a function field(t) of discriminant degree 12nwith a 2-torsion point forn=1,2,3, and with a 3-torsion point forn=1,2. For eachm=2,3, we parametrize the set of triples (E,P,T) of an elliptic curveE/with a rational pointPandm-torsion pointTthat satisfy certain integrality conditions by an open subset of2. We recover explicit equations for all elliptic surfaces (E,P,T) attaining each minimum by locating them as curves in our projective models. We also prove that forn=1,2 , these heights are minimal for elliptic curves over a function field of any genus. In each case, the optimal (E,P,T) are characterized by their patterns of integral points.

Generalized Artin'S Conjecture for Primitive Roots and Cyclicity Mod of Elliptic Curves Over Function Fields

1995 ◽  
Vol 38 (2) ◽  
pp. 167-173 ◽  
Author(s):  
David A. Clark ◽  
Masato Kuwata
Keyword(s):  
Finite Field ◽  
Prime Ideal ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Function Field ◽  
Function Fields ◽  
Artin's Conjecture ◽  
Primitive Roots

AbstractLet k = Fq be a finite field of characteristic p with q elements and let K be a function field of one variable over k. Consider an elliptic curve E defined over K. We determine how often the reduction of this elliptic curve to a prime ideal is cyclic. This is done by generalizing a result of Bilharz to a more general form of Artin's primitive roots problem formulated by R. Murty.

scholarly journals Rationality of dynamical canonical height

2018 ◽  
Vol 39 (9) ◽  
pp. 2507-2540
Author(s):  
LAURA DE MARCO ◽  
DRAGOS GHIOCA
Keyword(s):  
Elliptic Curve ◽  
Function Field ◽  
Rational Maps ◽  
Puiseux Series ◽  
Quadratic Maps ◽  
Canonical Height ◽  
Complete Answer ◽  
Irrational Values ◽  
Invent Math

We present a dynamical proof of the well-known fact that the Néron–Tate canonical height (and its local counterpart) takes rational values at points of an elliptic curve over a function field $k=\mathbb{C}(X)$, where $X$ is a curve. More generally, we investigate the mechanism by which the local canonical height for a map $f:\mathbb{P}^{1}\rightarrow \mathbb{P}^{1}$ defined over a function field $k$ can take irrational values (at points in a local completion of $k$), providing examples in all degrees $\deg f\geq 2$. Building on Kiwi’s classification of non-archimedean Julia sets for quadratic maps [Puiseux series dynamics of quadratic rational maps. Israel J. Math.201 (2014), 631–700], we give a complete answer in degree 2 characterizing the existence of points with irrational local canonical heights. As an application we prove that if the heights $\widehat{h}_{f}(a),\widehat{h}_{g}(b)$ are rational and positive, for maps $f$ and $g$ of multiplicatively independent degrees and points $a,b\in \mathbb{P}^{1}(\bar{k})$, then the orbits $\{f^{n}(a)\}_{n\geq 0}$ and $\{g^{m}(b)\}_{m\geq 0}$ intersect in at most finitely many points, complementing the results of Ghioca et al [Intersections of polynomials orbits, and a dynamical Mordell–Lang conjecture. Invent. Math.171 (2) (2008), 463–483].

scholarly journals Function Field Sieve Method for Discrete Logarithms over Finite Fields

1999 ◽  
Vol 151 (1-2) ◽  
pp. 5-16 ◽  
Author(s):  
Leonard M. Adleman ◽  
Ming-Deh A. Huang
Keyword(s):  
Finite Fields ◽  
Function Field ◽  
Sieve Method ◽  
Discrete Logarithms
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